Narrow Results

Given any amalgam $[S_1,S_2;U]$ of inverse semigroups, we show how to construct an amalgam $[T_1,T_2;Z]$ such that $S_1{\ast }_U{S}_2$ is embedded into $T_1{\ast }_Z{T}_2$, where $S_1\subseteq T_1$, $S_2\subseteq T_2$, $S_1\cap Z=S_2\cap Z=U$ and, for any $z\in Z$ and $h\in E(T_i)$ with $z\ge h$ in $T_i$, where $i\in \{1,2\}$, there exists $f\in E(Z)$ with $z\ge f\ge h$ in $T_i$; that is, $Z$ is a lower bounded subsemigroup of $T_1$ and $T_2$. A recent paper by the author describes the Schützenberger automata of $T_1{\ast }_Z{T}_2$, for an amalgam $[T_1,T_2;Z]$ where $Z$ is lower bounded in $T_1$ and $T_2$, giving conditions for $T_1{\ast }_Z{T}_2$ to have decidable word problem. Thus we can study $S_1{\ast }_U{S}_2$ by considering $T_1{\ast }_Z{T}_2$. As an example, we generalize results by Cherubini, Jajcayová, Meakin, Piochi and Rodaro on amalgams of finite inverse semigroups.

We investigate the connections between amalgams and Yamamura's HNN-extensions of inverse semigroups. In particular, we prove that amalgams of inverse semigroups with an identity adjoint are quotient semigroups of some special Yamamura's HNN-extensions. As a consequence, we show how to obtain the Schützenberger graph of a word w with respect to the presentation of an amalgamated free product starting from the Schützenberger graph of a suitable word w′ with respect to the presentation of the associated HNN-extension by simply making V-quotients and deletions of edges.